We found a match
Your institution may have rights to this item. Sign in to continue.
- Title
Persistent Cohomology and Circular Coordinates.
- Authors
Silva, Vin; Morozov, Dmitriy; Vejdemo-Johansson, Mikael
- Abstract
Nonlinear dimensionality reduction (NLDR) algorithms such as Isomap, LLE, and Laplacian Eigenmaps address the problem of representing high-dimensional nonlinear data in terms of low-dimensional coordinates which represent the intrinsic structure of the data. This paradigm incorporates the assumption that real-valued coordinates provide a rich enough class of functions to represent the data faithfully and efficiently. On the other hand, there are simple structures which challenge this assumption: the circle, for example, is one-dimensional, but its faithful representation requires two real coordinates. In this work, we present a strategy for constructing circle-valued functions on a statistical data set. We develop a machinery of persistent cohomology to identify candidates for significant circle-structures in the data, and we use harmonic smoothing and integration to obtain the circle-valued coordinate functions themselves. We suggest that this enriched class of coordinate functions permits a precise NLDR analysis of a broader range of realistic data sets.
- Subjects
OPERATIONS (Algebraic topology); LAPLACIAN operator; ALGORITHMS; SMOOTHING (Numerical analysis); DATA analysis
- Publication
Discrete & Computational Geometry, 2011, Vol 45, Issue 4, p737
- ISSN
0179-5376
- Publication type
Article
- DOI
10.1007/s00454-011-9344-x